- Apr 17, 2013Dear Randy,

Yes, S_{AB} is shorthand for (S_A)(S_B) etc. I always use barycentric coordinates unless the context clearly favors trilinear coordinates.

Let's check with the trilinear coordinates you gave. With an extra factor a (in the first component), the barycentric coordinates are

a cos^2 A/(cos A - cos B cos C) : ... : ...

= a(S_{AA}/(bc)^2) /(S_A/(bc) - S_{BC}/(a^2bc)) : ... : ...

= a^3S_{AA}/(a^2bc S_A - bcS_{BC}) : ... : ...

= a^4S_{AA}/(a^2S_A - S_{BC}) : ... : ...

Yes, it is the same as the one I gave!

Best regards

Sincerely

Paul

________________________________________

From: Hyacinthos@yahoogroups.com [Hyacinthos@yahoogroups.com] on behalf of rhutson2 [rhutson2@...]

Sent: Wednesday, April 17, 2013 3:37 PM

To: Hyacinthos@yahoogroups.com

Subject: [EMHL] Re: loci related to Taylor circle

Paul,

This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and as such would have trilinears (cos^2 A)/(cos A - cos B cos C) : :.

What do you mean by the notations S_{AB}, etc.? I assume it has to do with Conway notation. Also, are your coordinates trilinears or barycentrics?

Randy

--- In Hyacinthos@yahoogroups.com, "yiuatfauedu" <yiu@...> wrote:

>

> Dear Randy and Bernard,

>

> [RH] Let ABC be a triangle, and P a point.

> Let A'B'C' be the pedal triangle of P.

> Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.

> Define Cb, Ab, Ac, Bc cyclically.

> What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.

>

> [BG]: a quintic with many simple points but only two (I think) ETC centers : X4, X1498.

>

> *** X(1498) is the Nagel point of the tangential triangle.

> Ab = (0 : S_{AB}+S_{AC}-S_{BC} : S_{AB}+S_{AC}+S_{BC}) and

> Ac = (0 : S_{AB}+S_{AC}+S_{BC} : S_{AB}+S_{AC}+S_{BC})

> are isotomic points on BC, so are Bc, Ba, and Ca, Cb.

> X(1498) is the unique point with this property. The conic is

>

> (4/S^2)cyclic sum ((a^4S_{AA})/(S_{AB}+S_AC}-S_{BC}))yz - (x+y+z)^2 = 0,

> concentric (and homothetic) with the circumconic with perspector

> ((a^4S_{AA}/(S_{AB}+S_{AC}-S_{BC}):...:...)

> [with (6-9-13)-search number 5.10435062529...]

> and has center

> (a^4(S_{AAAB}+S_{AAAC}+S_{AABB}-S_{AABC}+S_{AACC}-S_{BBCC}/

> (S_{AB}+S_{AC}-S_{BC}) :...:...)

> with (6-9-13) search number 1.09478783248....

>

> Best regards

> Sincerely

> Paul

>

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